SUO: Re: Examples! Examples! Examples!
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
EEE. Note 14
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Example 1. John Sowa's "Top Level Categories" (cont.)
Solvitur somnambulando (it is solved by sleep-walking),
but for a spell so long solely as a sleep walk can run.
It strikes me and all my innominate funny bones how the
mind's eye can be so besooted with sclerotic motes that
it can hardly fathom what's lying smack dab in front of
the mind's face, right up to, and under the mind's nose.
And yet I must leave that sleeping hemisphere to sleep on it,
earnestly awaiting the outcoma of its long longed-for waking,
and return to some of the wool that I previously gathered on
the TLC example, reweaving through the text the mutations of
notation that are enscribed to the palimpsest of adaptations.
One "topic of pressingly obvious significance" (TOPOS) is the dimension or
distinction that stretches between models and theories, especially insofar
as it strikes me that I have yet to broach the lattice of theories that is
generated by the alphabet !TLC!.
And so to that task I now turn, with this review and revision:
The "space of models", or more precisely, a coordinate chart over the space
of models, is the universe of discourse TLC% = [!TLC!] = [a_1, ..., a_25].
This lays the maps of TLC^ = (TLC -> B) ~=~ Pow(TLC) ~=~ (B^25 -> B) over
the coordinate positions of TLC = <|a_1, ..., a_25|> ~=~ B^25, and so it
is assigned the "stereo type" TLC% : (TLC, TLC^) ~=~ (B^25, (B^25 -> B))
abbreviated as TLC% : (TLC +-> B) ~=~ (B^25 +-> B), or more briefly as
TLC% : [TLC] ~=~ [B^25].
A synopsis of these notations and further discussion can be found here:
D01. http://suo.ieee.org/ontology/msg04799.html
D02. http://suo.ieee.org/ontology/msg04800.html
D03. http://suo.ieee.org/ontology/msg04801.html
The "syntactic space" is the formal language L = L(!TLC!) c (!TLC! |_| !M!)*
of grammatical strings in the Cactus Language that we may variously describe
as "expressions", "formulas", "sentences", "terms", "wffs", or whatever fits.
A "zeroth order theory" about TLC is a set of sentences $T$ c L = L(!TLC!),
and thus there is a "lattice at large" of theories in the medium of !TLC!,
namely, (Pow(L), c).
So we have for starters at least two lattices:
1. One model lattice is (Pow(TLC), c), ordered by set-theoretic inclusion
or the "contained as a subset" relation that is here symbolized by "c".
This is isomorphic to the proposition lattice (TLC^, =>) that consists
of the propositions in (TLC -> B), ordered by the logical implication
relation that is here symbolized by "=>".
2. One theory lattice is (Pow(L), c), ordered by inclusion as usual.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o